Schrodinger estimates

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Schrodinger estimates

Solutions to the linear Schrodinger equation and its perturbations are either estimated in mixed space-time norms or , or in spaces, defined by

Note that these spaces are not invariant under conjugation.

Linear space-time estimates in which the space norm is evaluated first are known as Strichartz estimates. They are useful for NLS without derivatives, but are much less useful for derivative non-linearities. Other linear estimates include smoothing estimates and maximal function estimates. The spaces are used primarily for bilinear estimates, although more recently multilinear estimates have begun to appear. These spaces and estimates first appear in the context of the Schrodinger equation in [Bo1993], although the analogous spaces for the wave equation appeared earlier [RaRe1982], [Be1983] in the context of propogation of singularities. See also [Bo1993b], [KlMa1993].

Schrodinger Linear estimates

[More references needed here!]

On :

  • If , then
    • (Energy estimate)
    • (Strichartz estimates) references:Sz1997 Sz1977.
      • More generally, f is in whenever , and
        • The endpoint is true for KeTa1998. When it fails even in the BMO case Mo1998, although it still is true for radial functions Ta2000b, [Stv-p].In fact the estimates are true assuming for non-radial functions some additional regularity in the angular variable Ta2000b, although there is a limit as to low little regularity one can impose [MacNkrNaOz-p].
        • In the radial case there are additional weighted smoothing estimates available Vi2001
        • When one also has
        • When one can refine the assumption on the data in rather technical ways on the Fourier side, see e.g. VaVe2001.
        • When the estimate has a maximizer [Kz-p2].This maximizer is in fact given by Gaussian beams, with a constant of [Fc-p4].Similarly when with the estimate, which is also given by Gaussian beams with a constant of
    • (Kato estimates) Sl1987, Ve1988
      • When one can improve this to
    • (Maximal function estimates) In all dimensions one has for all
      • When one also has
      • When one also has The can be raised to TaVa2000b, with the corresponding loss in the exponent dictated by scaling. Improvements are certainly possible.
    • Variants of some of these estimates exist for manifolds, see [BuGdTz-p]
  • Fixed time estimates for free solutions:
    • (Energy estimate) If , then is also .
    • (Decay estimate) If , then has an norm of
    • Interpolants between these two are very useful for proving Strichartz estimates and obtaining scattering.

On T:

  • embds into Bo1993 (see also HimMis2001).
  • embeds into Bo1993. One cannot remove the from the exponent, however it is conjectured in Bo1993 that one might be able to embed into

On :

  • When embeds into (this is essentially in Bo1993)
    • The endpoint is probably false in every dimension.

Strichartz estimates are also available on [#manifold more general manifolds], and in the [#potential presence of a potential].Inhomogeneous estimates are also available off

the line of duality; see [Fc-p2] for a discussion.

Schrodinger Bilinear Estimates

  • On R2 we have the bilinear Strichartz estimate Bo1999:

and BkOgPo1998

Also, if u has frequency and v has frequency then we have (see e.g. [CoKeStTkTa-p4])

and similarly for .

  • The s indices on the right cannot be lowered, but perhaps the s indices on the left can be raised in analogy with the R2 estimates. The analogues on are also known KnPoVe1996b:

Schrodinger Trilinear estimates

  • On R we have the following refinement to the Strichartz inequality [Gr-p2]:

Schrodinger Multilinear estimates

  • In R2 we have the variant

where each factor is allowed to be conjugated if desired. See St1997b, references:CoDeKnSt-p CoDeKnSt-p.